Abstract
We present a new method, called the minimum cross-entropy (MCE) method for approximating the optimal solution of NP-hard combinatorial optimization problems and rare-event probability estimation, which can be viewed as an alternative to the standard cross entropy (CE) method. The MCE method presents a generic adaptive stochastic version of Kull-back’s classic MinxEnt method. We discuss its similarities and differences with the standard cross-entropy (CE) method and prove its convergence. We show numerically that MCE is a little more accurate than CE, but at the same time a little slower than CE. We also present a new method for trajectory generation for TSP and some related problems. We finally give some numerical results using MCE for rare-events probability estimation for simple static models, the maximal cut problem and the TSP, and point out some new areas of possible applications.
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References
E. H. L. Aarts and J. H. M. Korst, Simulated Annealing and Boltzmann Machines, John Wiley & Sons, 1989.
T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, Inc, 1991.
P. T. de Boer, D. P. Kroese, S. Mannor, and R. Y. Rubinstein, A Tutorial on the Cross-Entropy Method, Annals of Operations Research, 2005, (to appear).
D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, 1989.
J. N. Kapur, and H. K. Kesavan, Entropy Optimization with, Applications, Academic Press, Inc., 1992.
J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer: Berlin, Heidelberg, New York, 2001.
R. Y. Rubinstein, “The cross-entropy method for combinatorial and continuous optimization,” Methodology and Computing in Applied Probability vol. 2, pp. 127–190, 1999.
R. Y. Rubinstein, “Cross-entropy and rare event formula-native maximal cul and bipartition problems,” ACM Transactions on Modelling and Computer Simulation vol. 12(1) pp. 27–53, 2002.
R. Y. Rubinstein, and D. P. Kroese, The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning, Springer: Berlin, Heidelberg, New York, 2004.
R. Y. Rubinstein and B. Melamed, Modern Simulation and Modeling, John Wiley & Sons, Inc., 1998.
H. D. Wolpert, Information Theory—The Bridge Connecting Bounded Rational Game Theory and Statistical Physics. Manuscript, NASA Ames Research Center, in press.
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AMS 2000 Subject Classification: 65C05, 60C05, 68W20, 90C59
*This reseach was supported by the Israel Science Foundation (grant no 191-565).
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Rubinstein, R.Y. A Stochastic Minimum Cross-Entropy Method for Combinatorial Optimization and Rare-event Estimation*. Methodol Comput Appl Probab 7, 5–50 (2005). https://doi.org/10.1007/s11009-005-6653-7
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DOI: https://doi.org/10.1007/s11009-005-6653-7